Soft robotics possesses expansive utilization prospects in various domains, including medical aid, interior services, food and agriculture, and even space exploration. However, due to the limitations presented by the volume and weight of the driving, sensing and control systems, many soft robots are forced to rely on electrical or pneumatic tethers to transmit signals and supply energy, restricting their adaptability to structured environments. Fortunately, advancements in soft robotics have addressed this issue by integrating separate systems into a whole, achieving controllable and predictable autonomous motion. This paper reviews current integrated soft robots from bionic-inspired and function-oriented design inspirations, as well as their application perspectives. Integrated electrical, chemical, pneumatic and mechanical systems present numerous integrated design challenges and, therefore, reliable solutions to facilitate integration and intelligence are summarized herein from the perspectives of materials, fabrication, power source, sensing and control. Lastly, we discuss the potential of machine learning in solving current problems faced by soft robots and improving their behavioral intelligence.